## About the course

MAT 1362 is an introduction to rigorous mathematical reasoning and the concept of proof in mathematics. It is a course designed to prepare students for upper-level proof-based mathematics courses. We will discuss the axiomatic method and various methods of proof (proof by contradiction, mathematical induction, etc.). We do this in the context of some fundamental notions in mathematics that are crucial for upper-level mathematics courses. These include the basics of naive set theory, functions, and relations. We will also discuss, in a precise manner, the integers, integers modulo *n*, rational numbers, and real numbers. In addition, we will discuss concepts related to the real line, such as completeness, supremum, and the precise definition of a limit. In general, this course is designed to provide students with a solid theoretical foundation upon which to build in subsequent courses.

## Who should take this course?

MAT 1362 is a required course for several programs, including the honours and major programs in mathematics/statistics, the honours program in financial mathematics and economics, and the joint honours programs of mathematics with economics and physics

In addition to students in the above programs, this course may be of interest to students who feel dissatisfied with mathematics courses in which computation is favoured over proof and explanation. In MAT 1362, we will rarely perform computations based on standard techniques and algorithms, such as computing derivatives using the chain and product rules. Instead, we will focus on **why** certain mathematical statements are true and **why** given techniques work. This is often much more difficult than following an algorithm to perform a computation. However, it is much more rewarding and results in a deeper understanding of the material.

## Official course description

Elements of logic, set theory, functions, equivalence relations and cardinality. Proof techniques. Concepts are introduced using sets of integers, integers modulo *n*, rational, real and complex numbers. Exploration of the real line: completeness, supremum, sequences and limits. Some of the concepts will be illustrated with examples from geometry, algebra and number theory.

## Prerequisites

MAT 1339 or Ontario 4U Calculus and Vectors (MCV4U) or an equivalent. MAT 1362 and MAT 1348 cannot be combined for credits.

## Reference material

### Course text

Matthias Beck and Ross Geoghegan, The Art of Proof, 2010. ISBN: 978-1-4419-7022-0 (Hardcover) 978-1-4419-7023-7 (eBook).

While on campus (connected to the university network), this book is available for free download by clicking on the link above.

### Lecture notes

Click the above header to download lecture notes for the course. They are a single file that will be updated as the course progresses. They should be complemented by your own class notes. I will often say more in class than is in the lecture notes.

## DGDs

You should be registered in one of the two DGDs associated to this course. The DGDs are an important part of the course. They are led by a graduate student and supplement the material covered in the lectures. You will be given an opportunity to ask questions about the lectures. In addition, the TA will go through the recommended exercises that correspond to the lectures from the past week. To get the most from the DGDs, you **should attempt the recommended exercises before the DGDs**. That way, you can focus on clearing up any difficulties that you encounter. Graded homework assignments and midterm tests will also be handed back in the DGDs.

## Extra help

The Math Help Centre does not serve this course. However, if you are looking for extra help beyond the DGDs and office hours, you can hire a private tutor using the tutor referral service of the Peer Help Centre.

## Course website

There are two main locations for information regarding the course:

- alistairsavage.ca/mat1362/. This is the main course website. It will be updated regularly and contains important material for the course.
- Virtual Campus. Grades will be posted on Virtual Campus.